In my class on Elliptic Curves, we showed that in a Projective Plane a line intersects an elliptic curve at exactly 3 points including point at infinity (with multiplicities) satisfying Bezout's Theorem.
But the point at infinity, $\mathcal{O}$, does not need to be $(0,1,0)$. It can be any point $(x,y,z)$ such that $z=0$.
So, now if I have a line intersecting 3 points of the curve (say $P,Q,R$), I can let my $\mathcal{O}$ be in the direction of the line segment $\overline{PR}$ and then the full line will intersect the curve at a $4^{\text{th}}$ point, which is $\mathcal{O}$ but this should not be possible as per Bezout's Theorem.
So, how does Bezout's Theorem hold when $\mathcal{O} \neq (0,1,0)$?